Awesome! It pushed correctly. However, I am a SUPER newbie to this, and I'm not sure what to do after I've finished the ADB part of it. On the phone, do I need to install it? Again, I checked in the Market Downloads and under Manage Applications (as well as Live Wallpapers, just in-case), and I don't see anything. I know this is gonna be something simple, I just don't quite know what I'm doing yet.

That's because "sysrw" is not an adb command. It is usually a .sh script which contains the longer command that some developers have included in their roms to be run from a terminal on the phone. However while in adb shell a "adb remount" usually does the same thing.

I know this is a necro thread but I thought I would chime in for those that need help since this comes up as the top on google search for "Read-Only ADB push".

If you're running into this:
1. Make sure you are rooted with SuperUser of some type installed
2. On your machine with adb installed, do the following:
[HIGH]
--# adb devices
(make sure your device shows up)
--# adb root
(this restarts the adb daemon as root on the phone)
--# adb push blah bleh
(this should now run without issue)
[/HIGH]
You DO NOT need to remount the /system symlink with rw permissions. A lot of people will do this and forget to remount it back to ro permissions.